L11n196

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_12,3,13,4 X_13,21,14,20 X_16,9,17,10 X_10,19,11,20 X_15,7,16,22 X_21,15,22,14 X_18,5,19,6 X₂₇₃₈ X_4,11,5,12 X_6,17,1,18
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 4, -5, 10, -2, -3, 7, -6, -4, 11, -8, 5, 3, -7, 6}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{2}v^{3}-3u^{2}v^{2}+4u^{2}v-u^{2}+uv^{4}-4uv^{3}+7uv^{2}-4uv+u-v^{4}+4v^{3}-3v^{2}+v}{uv^{2}}}$ (db)
Jones polynomial	$q^{5/2}-4q^{3/2}+7{\sqrt {q}}-{\frac {10}{\sqrt {q}}}+{\frac {12}{q^{3/2}}}-{\frac {12}{q^{5/2}}}+{\frac {10}{q^{7/2}}}-{\frac {8}{q^{9/2}}}+{\frac {4}{q^{11/2}}}-{\frac {2}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}z^{-1}+a^{5}z^{3}-a^{5}z-a^{5}z^{-1}-a^{3}z^{5}-a^{3}z^{3}-a^{3}z-az^{5}-az^{3}+z^{3}a^{-1}-az$ (db)
Kauffman polynomial	$-a^{5}z^{9}-a^{3}z^{9}-a^{6}z^{8}-5a^{4}z^{8}-4a^{2}z^{8}-a^{5}z^{7}-8a^{3}z^{7}-7az^{7}+5a^{4}z^{6}-2a^{2}z^{6}-7z^{6}-3a^{7}z^{5}+2a^{5}z^{5}+16a^{3}z^{5}+7az^{5}-4z^{5}a^{-1}+2a^{6}z^{4}+5a^{4}z^{4}+12a^{2}z^{4}-z^{4}a^{-2}+8z^{4}+8a^{7}z^{3}+4a^{5}z^{3}-7a^{3}z^{3}+3z^{3}a^{-1}+a^{6}z^{2}-4a^{4}z^{2}-7a^{2}z^{2}-2z^{2}-6a^{7}z-4a^{5}z+a^{3}z-az-a^{6}+a^{7}z^{-1}+a^{5}z^{-1}$ (db)

Khovanov Homology

The coefficients of the monomials

t^{r}q^{j}

are shown, along with their alternating sums

\chi

(fixed

j

, alternation over

r

).

\		r
	\
j		\

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

6

1

-1

4

3

2

4

1

-3

0

6

3

-2

7

5

-2

-4

5

0

-6

5

7

2

-8

3

5

-2

-10

1

5

4

-12

1

3

-2

-14

2

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-6$	${\mathbb {Z} }^{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-2$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-1$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=0$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{6}$
$r=1$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=3$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$